Actually, for most of its existence, Cyrillic has had a θ (theta) like in Greek, used only in loanwords and pronounced either as ф (f) or т (t) because the th sound is not part of Slavic phonetics. θ was dropped fairly recently - in the 20th century.
It absolutely doesn't as pronounced now, yet Thomas is Фома, Theodor is Феодор, etc. Just like Hertz is Герц, even through Г and H are as far from each other as one can get.
https://old.reddit.com/r/conorthography/top/
Or you can take the plunge and make your own complete script:
https://old.reddit.com/r/neography/top/
And if you're still not satisfied and want to spend the rest of your life making a language that no one else will care to learn:
https://old.reddit.com/r/conlangs/